# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 35 (2006) pp.767-778

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### Abstract

In this note, we consider a maximal operator $\sup_{t \in \mathbb{R}}|u(x,t)| = \sup_{t \in \mathbb{R}}|e^{it\Omega(D)}f(x)|$, where $u$ is the solution to the initial value problem $u_t = i\Omega(D)u$, $u(0) = f$ for a $C^2$ function $\Omega$ with some growth rate at infinity. We prove that the operator $\sup_{t \in \mathbb{R}}|u(x,t)|$ has a mapping property from a fractional Sobolev space $H^\fraca{1}{4}$ with additional angular regularity in which the data lives to $L^2((1 + |x|)^{-b}dx) (b > 1)$ . This mapping property implies the almost everywhere convergence of $u(x,t)$ to $f$ as $t \to 0$, if the data $f$ has an angular regularity as well as $H^\frac{1}{4}$ regularity.

MSC(Primary) 42A45 42B25 Schr\"{o}dinger type equation, maximal operator, angular regularity